The proof of Lemma REF  is an adaptation of the arguments of [1]} to our situation of functional depending on set-function pair with extra care paid for the constraint on the number of boundary connected components. The idea is to treat the boundary of each admissible region as a jump of a properly defined displacement. In particular, we choose such displacement of the type (REF ), where \(\xi \)  is selected as in the construction used in the proof of [2]}.
